Download The Liar Paradox in BASIC language

The Liar Paradox in BASIC language
(Symbolic Logic and Automata)
Every truth function corresponds to an isomorphic digital circuit. Consequently, the logical
structure of every proposition can be presented within the range of propositional logic as an
equivalent digital circuit. Providing that the logical values ‚true‘ and ‚false‘ correspond to the
‚high‘ and ‚low‘ voltage levels, the output of a circuit being equivalent with contradiction is
always low level for every input state, while the output of a circuit corresponding to a
tautology is always high level, irrespective of the input states. On the other hand, remaining
propositions correspond to circuits the output of which is high level if and only if (hence iff)
some of the atomic components of the proposition are true, or rather, the inputs equivalent
with the atomic propositions are high level. But what is the equivalent of a circulating
statement or argumentation? The propositions are true or false irrespective of time, whereas
the voltage level of the circuits can change in time. To be more precise, we can say that the
input levels of the circuits are high or low depending on whether we evaluate the atomic
formulae of the formula expressing the logical structure of proposition true or false. The
voltage level of the output of the circuit and truth-value of the formula result from it. I will
call ‚combinational automaton‘ the digital circuits, which thus may model the formulae of
propositional calculus. Since one and the same atomic formulae can compose more complex
formulae, one automaton can have more outputs, where every output corresponds to a
complex formula.
Formulae connected with truth functions yield formulae again. Although there are always
corresponding automata for them, the situation is not as simple in this case. We don’t get
combined automata joined to each other in each case, and it is also possible that we don’t get
an automaton – operating machine or circuit – at all. To avoid this, it is sufficient to comply
with the following rules:
(a) It is permitted to join any two inputs of any two automata.
(b) Arrange the automata on neighbouring levels and join the output of an automaton only
with the input of another on higher level.
At every point of time, the output state of a combinational automaton is unambiguously
determined by the state of inputs. In other words, the output of the machine is the function of
its input. The two rules given above will guarantee this. If this were not case, it would not be
possible to simulate the automaton by logical formulae, since the truth value of a logical
formula is unambiguously determined by the truth value of its atomic formulae, and,
correspondingly, the truth value of a proposition is a function of the truth value of its
components. Therefore, the rules given in (a) and (b) are interpreted in the world of
propositions in the following way: the truth-value of a proposition can never be influenced by
itself or by the truth-value of its components. (Though, it my be influenced by other
attributes.) There are digital circuits the output of which is not a function of the input. These
are not combinational automata but they can be constructed from components that are
combinational automata themselves. They can be constructed for instance NAND gates
alone, operating the following way: both input and output are either on high level or low level
at any point of time; the output is low level at a point of time iff at the same time both inputs
are high level. If we connect the two inputs we get a new simple circuit, the ‚inverter‘. The
output of the inverter is high if the input is low and inversely: the output level is low of the
input level is high. We can consider the inverter as the presentation of the negation if the
logical values ‚true‘ and ‚false‘ correspond to the ‚high‘ and ‚low‘ levels respectively. E.g.
the input level corresponds to a valuation of the formula ‚p‘ and the output level corresponds
to the values of the formula ‚~p‘. Its operation will be given in the formula below. Though
this will not reflect the causal relation between the input and output, it will suit out present
purposes.
1=V(in.t)  0=V(out,t)
0=V(in,t)  1=V(out,t)
The graphic illustration of the inverter is given in Fig. 1:
The fact that an x input of an automaton is connected with another y input or output can be
expressed as x=y.
The general condition of the connectability of an input with an output or another input is
given in condition (c).
c. The x input or output of A1 automaton can be connected with y input or output of A2
automaton iff the formulae describing the operation of the two automata together with the
describing of connection can not lead to a contradiction. Here is an example when the
condition is not fulfilled Fig. 2:
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The figure does not show an automaton because its description leads to contradiction.
Namely, if x is high level the y is low level but x=y, therefore x ought to be low level, which
is contradiction.
The range of automata restricted by (c) is wider than that of the combinational automata. It
includes machines the input states of which don’t determine unambiguously the output states,
i.e. the output is not a function of the input. Most digital circuits belong to this latter group. I
will call them ‚sequential automata‘. Figure 3. is an illustration of such an automaton. The
diagram shows the output signal of the machine.
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The unit named DELAY operates in which a way that its output state in every t period is
equivalent with the output state of the previous period (t-1). Therefore, the automaton gives
alternating high level and low-level signals in the subsequent periods. This is how electric
bell works. The arm of the bell draws in only if there is electric power is the coil, and
conversely, electric power can run through the coil if the arm is not drown in. The description
of the automaton is given in the formulae below:
1=V(out,t-1) 0=V(out,t)
0=V(out,t-1) 1=V(out,t)
Neglecting the time, the description of the automaton can be simplified as:
1=V(out) 0=V(out)
0=V(out) 1=V(out)
We can denote the output of the automaton by a number. For example let it be -5. Then the
two formulae above have to be changed in this manner:
1=V(-5) 0=V(-5)
0=V(-5) 1=V(-5)
The question arises if there is a proposition corresponding to such an automaton. In my view,
there is one to be found, though not a proposition but a circulating statement. Let the names
of sentences be numbers. If x is the name of a sentence, then S$(x) is the sentence that is
denoted by x; if x is not the name of a sentence then S$(x) is the empty character. If a welldefined environment is given then the function V determines the values of the sentences. If x
and y are names of one and the same sentence then they have identical values. If the sentence
S$(x) in a well-defined environment expresses a proposition, then:
(A1) ~0=V(x)  1=V(x)
(A2) ~1=V(x)  0=V(x)
(A3) 1=V(x)  S$(x)
(A4) 0=V(x)  ~S$(x)
(A5) If p is a proposition and »p«=x, then pS$(x).
On the basis of all this an appropriate formula for the liar paradox can be given in the
following form:
(1) »0=V(-5)«=-5
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(2) The sentence denoted by -5 expresses a proposition.
The argumentation:
(3) 0=V(-5)  S$(-5)
(4) 1=V(-5)  S$(-5)
(5) S$(-5)  0=V(-5)
(6) 0=V(-5)  ~S$(-5)
(7) ~S$(-5)  ~0=V(-5)
(8) ~0=V(-5)  1=V(-5)
(1)(2)(A5)
(2)(A3)
(3)
(2)(A4)
(3)
(2)(A1)
Whence: 1=V(-5)  0=V(-5)
0=V(-5)  1=V(-5)
(4)(5)
(6)(7)(8)
In my view, logical structure of the argumentation coincides with the operation of the
automaton – Fig.3. the bell – irrespective of the logical correctness of the paradox itself. To
illustrate this, I give a formulation of the paradox in the popular BASIC language for
machine Commodore 64. The first line of the program clears the screen. The second half of
line 250 means the motion of the cursor upward twice, but if p=1 then four times. Lines from
150 to 156 check your formula. Apart from the technical details, the program is similar to the
argumentation. In such a way the operation of the computer shows the truth-value of the
paradox. Here is the pudding for you to be tested.
....
Ferenc András
This version was written in 1985
[email protected]
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